Before us is Mishnah 8 in the fifth chapter of Maseches Pesachim. The Mishnah establishes that everything done for the korban Pesach on a weekday was done in exactly the same way on Shabbos as well, with no difference at all, except for one thing: "shehakohanim medichin es ha'azarah shelo kirtzon chachamim" - on Shabbos the kohanim would rinse down the azarah, the courtyard of the Beis HaMikdash, and this was done contrary to the will of the Sages, who objected to this activity. This activity was of course done on weekdays as well; it was only on Shabbos that it was against the will of the Sages.
The reason for the Sages' objection:
The Sages held that there is a shevus here, a rabbinic prohibition, based on the halachah that one may not wash floors on Shabbos. The reason is the concern of leveling depressions, that is, smoothing out dips and holes in the ground, which is a derivative of the melachah of plowing, a branch that emerges from it, according to which one may not improve the ground in any way. True, this concern is not relevant in the Beis HaMikdash, but this is the rabbinic prohibition that was established on the basis of other situations in which one may not wash the floor.
Although in general the rule is that "there is no shevus in the Mikdash," meaning that rabbinic prohibitions are not applied in the Beis HaMikdash, this depends on how necessary the activity is. The Sages held that it was not all that necessary, while the kohanim held that it was.
How the rinsing was done in the Mikdash:
The system in place in the Beis HaMikdash was that a water channel ran through the Mikdash at all times. In order to wash down the azarah, the kohanim would block the water's outlet, so that the water would fill up over the floor of the azarah and wash away the blood, and afterward they would open the outlet and the water would drain out. This was their practice, and the dispute revolved around doing this on Shabbos.
The words of Rabbi Yehudah - a cup from the mixed blood:
The Mishnah continues: "Rabbi Yehudah omer: kos hayah memaleh midam hata'aroves, uzerako zerikah achas al gabei hamizbe'ach" - while all the blood was lying on the floor, they would take a cup and fill it from the mixed blood on the ground, and throw it in a single throw against the mizbe'ach. The reason: perhaps there was a korban Pesach whose blood spilled from the vessel onto the floor, and if its blood was not thrown against the mizbe'ach, that korban is not valid. In order to ensure its validity, Rabbi Yehudah said that they should fill a cup from all the mixed blood and throw it against the mizbe'ach. However, the Mishnah concludes: "velo hodu lo chachamim" - the Sages did not agree with him.
The basis of the dispute: does blood nullify blood?
The root of the dispute lies in a fundamental question about which Rabbi Yehudah and the Sages disagreed - does blood nullify blood:
The view of the Sages: Blood does nullify blood, and a small quantity of blood is nullified in a larger quantity. The blood on the floor is generally dam hatamtzis, the blood that seeps out afterward, which is not dam hanefesh - it is not the blood whose exit caused the animal's death, and it is not the blood of the initial slaughter. Since it is not that initial blood, it is not fit and it is not meant to be thrown on the mizbe'ach. Consequently there is no benefit at all in this throwing, since the blood of the shechitah that fell in and was not thrown is nullified.
The view of Rabbi Yehudah: Blood does not nullify blood, and even the tiniest quantity is not nullified. Therefore there is benefit and purpose in throwing the mixture, since that tiny quantity of blood that was not thrown in time is now thrown and becomes valid. For this reason he held that the blood should be thrown on the mizbe'ach.
In summary: In this Mishnah we learned that the order of the Pesach service on Shabbos was identical to its order on a weekday, except for the rinsing of the azarah, which was done against the will of the Sages because of the shevus of leveling depressions; we clarified how the rinsing was carried out by blocking the water channel and then opening it; and we learned the dispute between Rabbi Yehudah and the Sages regarding the cup of mixed blood, which is rooted in the question of whether blood nullifies blood.